Algebraic genericity within the class of sup-measurable functions
A function \(F:\mathbb{R}^2\to\mathbb{R}\) is Lebesgue (Baire) sup-measurable if, for each Lebesgue (Baire) measurable function \(f:\mathbb{R}\to\mathbb{R}\), the superposition \(F_f:\mathbb{R}\to\mathbb{R}\), \(F_f(x)=F(x,f(x))\), is Lebesgue (Baire) measurable. The problem of existence of non-measurable sup-measurable functions is undecidable in ZFC. Under assumption of CH (or some weaker set-theoretical assumptions) non-measurable sup-measurable functions were constructed: \(\bullet\) in the measure case by \textit{Z. Grande} and \textit{J. S. Lipinski} [Colloq. Math. 39, 77--79 (1978; Zbl 0386.28006)], and independently, by \textit{A. B. Kharazishvili} [Soobshch. Akad. Nauk Gruz. SSR 100, 533--536 (1980; Zbl 0454.28005)], see also [\textit{A. B. Kharazishvili}, J. Appl. Anal. 3, No. 2, 211--223 (1997; Zbl 0901.28002)]; \(\bullet\) in the category case by \textit{E. Grande} and \textit{Z. Grande} [Fundam. Math. 121, 199--211 (1984; Zbl 0573.28007)]. On the other hand, the model of ZFC without of non-measurable sup-measurale functions was constructed \(\bullet\) in the measure case by \textit{A. Rosłanowski} and \textit{S. Shelah} [Isr. J. Math. 151, 61--110 (2006; Zbl 1125.03036)]; \(\bullet\) in the category case by \textit{K. Ciesielski} and \textit{S. Shelah} [J. Appl. Anal. 6, No. 2, 159--172 (2000; Zbl 0972.03048)]. In this paper the authors show that under CH (or some weaker set-theoretical assumptions) there is a continuum-dimensional vector space contained in the family of all Lebesgue (Baire) non-measurable sup-measurable functions (plus the zero function). In the second part of the paper, the authors consider also the families of weakly sup-measurable functions with respect to measure or category. A function \(F:\mathbb{R}^2\to\mathbb{R}\) is weakly sup-measurable if, for every continuous, differentiable almost everywhere function \(f:\mathbb{R}\to\mathbb{R}\), the superposition \(F_f\) is measurable, see [\textit{A. B. Kharazishvili}, J. Appl. Anal. 3, No. 2, 211--223 (1997; Zbl 0901.28002)].
scientific article; zbMATH DE number 7138436
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